reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem Th57:
  uReal.r1 * uReal.r2 == uReal.(r1*r2)
proof
  uReal.r1 = Unique_No sReal.r1 & uReal.r2 = Unique_No sReal.r2 by Def7;
  then uReal.r1 * uReal.r2 == uReal.r1 * sReal.r2 ==
  sReal.r1 * sReal.r2 by SURREALO:def 10,SURREALR:51;
  then uReal.r1 * uReal.r2 == sReal.r1 * sReal.r2 ==
  sReal.(r1*r2) by Lm19,SURREALO:4;
  then A1:uReal.r1 * uReal.r2 == sReal.(r1*r2) by SURREALO:4;
  uReal.(r1*r2) = Unique_No sReal.(r1*r2) by Def7;
  then sReal.(r1*r2) == uReal.(r1*r2) by SURREALO:def 10;
  hence thesis by A1,SURREALO:4;
end;
